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Given a Hermitian line bundle $L$ over a flat torus $M$, a connection $\nabla$ on $L$, and a function $Q$ on $M$, one associates a Schrödinger operator acting on sections of $L$; its spectrum is denoted $Spec(Q;L,\nabla )$. Motivated by work of V. Guillemin in dimension two, we consider line bundles over tori of arbitrary even dimension with “translation invariant” connections $\nabla$, and we address the extent to which the spectrum $Spec(Q;L,\nabla )$ determines the potential $Q$. With a genericity condition, we show that if the connection is invariant under...

We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds , more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.